Refer to a circle with center O and radii OP and OR meeting the circle at P and R. What arc is subtended by central angle ∠POR?

In the same circle, if chords AB and DC are drawn and central angles ∠AOB and ∠COD intercept arcs AB and CD respectively, which equation is correct?

Given a circle with center O and points A, B, C, D on the circle such that AB = 40° of arc and CD = 60° of arc, find the central angle ∠AOC that intercepts arc AC when arc AC consists of arcs AB and BC and ∠BOC intercepts arc BC. Which value is correct if ∠BOC = 80°?

Which statement best defines an inscribed angle in a circle?

Consider a circle with center O. Points A and B are on the circle, and chord AN and BN meet at N on the circle, forming inscribed angle ∠ANB that intercepts arc AOB. Which relationship is correct for an inscribed angle?

In a circle, a chord-chord inscribed angle subtends arc x. If the measure of arc x is 80°, what is the measure of the inscribed angle that intercepts arc x?

A tangent y touches a circle at point A. The chord AX subtends arc x. What is the measure relationship between the angle formed by the tangent y and chord AX at A and arc x?

In a circle, an inscribed angle subtends arc AC. According to the inscribed angle theorem, which expression correctly gives the measure of ∠ABC?

Points A, B, C lie on a circle with center O. If central angle ∠AOC intercepts arc AC, what is the relationship between ∠AOC and the inscribed angle ∠ABC that subtends the same arc AC?

In a circle, angles ∠BAD and ∠DAC are inscribed and intercept arcs BD and DC respectively. Which equation shows how to compute ∠BAC using arcs BD and DC?

In triangle ABC inscribed in a circle, the measures of inscribed angles at A, B, C are 40°, 60°, and 80° respectively. Which is the measure of arc BC?

For the same triangle ABC inscribed in a circle with ∠A = 40°, ∠B = 60°, ∠C = 80°, what is the measure of arc AC?

Given a circle with center O and a diameter AB. Points P and Q lie on AB such that AP, AQ, BP, and BQ are drawn to points on the circle forming inscribed angles at A and B. Which statement is true about inscribed angles ∠APB, ∠AQB, and ∠ARB that subtend the same arc AB?

In a cyclic quadrilateral with both pairs of opposite vertices connected by equal arcs, which conclusion about the indicated inscribed angles a, b, c, d is consistent with the property 'inscribed angles subtending the same arc are equal'?

In the circle with points M, N, P, Q on the circumference, it is given that inscribed angle at M subtending arc PQ equals 32°, and angle at Q subtending arc MN equals 10°. Which statement correctly applies the same-arc property?

In circle ABCD, points A, B, C, D lie on the circle. Given BD = DC and angle ABD = 42°. If angle DBC = 65°, what is angle BDC?

Using the same configuration where BD = DC and angle DBC = 65°, which statement justifies that angle DCB = angle DBC?

With BD = DC and angle ABD = 42°, angle DBC = 65°, find angle BAC.

In the same figure, compute angle DAB.

Still referring to the configuration with BD = DC and angle ABD = 42°, angle DBC = 65°. Find angle DEC, where E is the intersection of diagonals AC and BD inside the circle.

A circle has center O and points A, B, C on the circle with chord AC. Which relationship expresses the central angle LAOC in terms of the inscribed angle LABC that subtend the same arc AC?

If the inscribed angle LABC equals 34°, what is the measure of the corresponding central angle LAOC that subtends the same arc AC?

In a circle, an inscribed angle that subtends the same arc as a central angle measures half the central angle. If the central angle is 76°, what is the measure of the inscribed angle x that intercepts the same arc?

In circle PQRS with center O, angle QPR is an inscribed angle measuring 100°. What is the measure of the central angle y that subtends the same arc QR?

In circle WXYZ with center O, the central angle XOY measures 130°. The reflex central angle WOY spans the remainder of the circle. If a denotes the inscribed angle subtending arc WY (the reflex arc), what is a?

In triangle formed by chords AB, BC, and AC of a circle with center O, suppose angle OBC = 30°. Points B and C lie on the circle with BO and CO as radii to chord BC. If OB and OC are equal, what is the measure of the central angle BOC?

In the same configuration as the previous question, the central angle BOC found equals 120°. What is the measure of the inscribed angle BAC that subtends arc BC?

In a circle with diameter AB and center O, point C lies on the circle. What is the measure of angle ACB?

In triangle ABC inscribed in a circle with AB as the diameter, angle A is 60°. What is the measure of angle B if C is on the circle?

In a circle with center O, chords AD and CB intersect at E inside the circle. If ∠AOB = 60°, which inscribed angle that subtends arc AB equals 30°?

A circle has diameter AB. From a point T outside the circle, TP is tangent at point P and TA is a secant through A. If ∠TAP = 35° and OT ⟂ TP, what is ∠APT?

In triangle ABC inscribed in a circle with AB as diameter, suppose ∠BAC = 21°. What is the measure of angle at B?

Which statement is always true for any triangle inscribed in a circle with AB as the diameter?

Which statement best defines a cyclic quadrilateral?

In any cyclic quadrilateral ABCD, what is always true about a pair of opposite angles?

If ABCD is a cyclic quadrilateral, which equation also holds true by the opposite angles theorem?

A cyclic quadrilateral ABCD is drawn so that arc BCD corresponds to angle ∠BAD at the circumference. Which formula from inscribed angle properties is used in the proof that opposite angles are supplementary?

In a cyclic quadrilateral, if ∠A = 80°, what is ∠C?

In a cyclic quadrilateral ABCD, ∠B = 70°. What is the measure of ∠D?

A diagram shows a cyclic quadrilateral with labeled interior angles a, b, c, d in order around the circle. Which pair must add to 180°?

In the worked example, a cyclic quadrilateral has ∠A = 80° and ∠B = 70°. What are the values of y = ∠DAB's opposite angle and x = ∠ADC's opposite angle?

In a cyclic quadrilateral, the exterior angle at a vertex equals which of the following?

Quadrilateral PQRS is cyclic. If an exterior angle at T formed by extending side PS equals z°, and the interior opposite angle at Q equals x°, what relationship holds?

In a cyclic quadrilateral ABCD, if ∠CDE = 70° is an exterior angle at D and ∠CDB = 72°, what is ∠ADB?

For the same configuration as above, using the property of a cyclic quadrilateral, what is ∠BCD?

Two intersecting circles overlap. Quadrilateral ABDC is cyclic in the left circle and CDFE is cyclic in the right circle. If ∠ABD = 85°, what is a = ∠ACD in the left circle?

In cyclic quadrilateral EFGH, diagonals meet at O. Given ∠DOH = 82° for points D, O, H on a circle and ∠DGE = 17° in the same figure, what is ∠HEG?

Using the same EFGH configuration, what is ∠HGF if ∠OGH = ∠OHG and ∠DGE = 17°, ∠EGF = 49°?

Continuing from the EFGH problem, what is ∠FEG if ∠FEH + ∠HGF = 180° and ∠HGF = 107° and ∠GEH = 49°?

In a cyclic quadrilateral ABCD, with AB = BC and CD extended to E, if ∠ACE = 27° and ∠DAE = 76°, what is ∠ACB?

Angles with vertex inside a circle: If point E is inside circle ABCD and AE and CE are chords such that ∠AED equals half the sum of intercepted arcs AD and BC, which formula matches this statement?

When two chords intersect at point E inside a circle, the measure of the vertical angles at E are related how?

Which statement is always true for a cyclic quadrilateral?

In a circle, a tangent at point A and a chord AB form an angle ∠TAB outside the circle at the point of tangency. According to the tangent–chord theorem, ∠TAB equals which interior angle?

A circle has tangent AT at A. Chord AB is drawn. If ∠BAD is an angle in semicircle ADB (with AD a diameter), what is ∠TAB?

In a circle with tangent ST at E, chords EF and EG are drawn with point H on arc EG. If ∠HES = 30° and the angle at the circumference intercepting arc GE near F is 56°, what is ∠GFE?

Which statement is the best expression of the tangent–chord theorem?

In circle O, chords AC and AB are equal. If inscribed angle ABC measures 65°, what is the measure of angle ACB?

In circle O, chords AC and AB are equal. If ∠ABC = 65°, what is the measure of central angle ABO (angle at O subtending arc AB)?

In the same figure of the cyclic quadrilateral from the previous question, what is the measure of angle BAE (angle subtended by arc BE at A)?

In the same figure of the cyclic quadrilateral from earlier, what is the measure of angle BDC?

In a circle, from a point outside, two secants make angles of 60° at a point on the circle and 34° at another, as shown. What is the measure of angle x at the exterior point?

In circle with center O, AC is a chord and BD is a chord through an interior point I on AC. Given ∠CAB = 40° and ∠AIB = 60°, find ∠ACD.

Using the same setup as the previous problem, what is the measure of angle CAD?

In circle O, triangle ABC is inscribed with central angle ∠AOC = 120°. What is the measure of inscribed angle ABC?

In circle O, chords AD and BC are parallel. Given ∠ACB = 45° and ∠ACD = 25°, find ∠ADC.

With the same configuration where AD ∥ BC, what is the measure of angle OAB (angle between OA and AB)?

Using the same tangent configuration, find ∠APC (angle formed at P by tangents PA and PC where PC is a secant through C).

Using the same configuration, find ∠DAB.

In the same figure, what is the measure of central angle BOC?

In a circle with center O, diameter RS and chord PQ form an angle at T on the circle where ∠ROQ = 48°. Which inscribed angle equals 24°?

In circle O with triangle ABC inscribed, BC is extended to point E and AC is extended to point D as shown. If inscribed angle at A is 110° and angle at B on the circle near BC is 40°, find x = ∠ACB.

In cyclic quadrilateral ABCD with AB = BC and ∠ACE = 25° where E is on the extension of CD, and ∠ADE = 80°, find ∠ABC.

In the same quadrilateral, find ∠BAD.

Two congruent circles with centers K and N intersect. Square JKLN is inscribed so that vertices J,K,L,N lie on the common circle arcs as shown. Given ∠TJS = 56° with T and S on one circle and ∠NJK = 40°, find the measure of ∠MLN.

With the same configuration, find ∠LMN.

With the same configuration, find ∠KLKN (angle at L subtended by chord KN).